- #1

Combinatorics

- 36

- 5

## Homework Statement

Let [tex] p [/tex] be prime and let [tex] b_1 ,...,b_k [/tex] be non-negative integers. Show that if :

[tex] G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k} [/tex]

then the integers [tex] b_i [/tex] are uniquely determined by G . (Hint: consider the kernel of the homomorphism [tex] f_i :G \to G [/tex] that is multiplication by [tex] p^i [/tex] . Show that [tex] f_1 , f_2 [/tex] determine [tex] b_1[/tex].

Proceed similarly )

## Homework Equations

The question, together with the theorem it should help me prove is attached (this question should help me prove the uniqueness part of the theorem)

## The Attempt at a Solution

I've tried using the hint, and considering the homomorphism [tex] f_1 : G \to G [/tex] that is defined by [tex]f_1 (x ) = px [/tex] . Its kernel is [tex] ker f_1 = (\mathbb{Z}_p ) ^{b_1} \bigoplus (\mathbb{Z}_p) ^ {b_2} ... [/tex].

But how does it help me? Am I right in my calculation of the kernel? Hope someone will be able to help me

Thanks in advance !